Question: Find one value of $x$ that is a solution to the equation: $(3x+5)^2+5(3x+5)+6=0$ $x=$
We could solve for $x$ by expanding $(3x+5)^2$ and $5(3x+5)$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that if we let ${p}={3x+5}$, we can rewrite the equation: $({3x+5})^2+5({3x+5})+6=0$ In particular, we can express it in the form: ${p}^2+5{p}+6=0$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2+5{p}+6&=0\\\\ ({p}+3)({p}+2)&=0\\\\ {p}=-3\ &\text{or} \ \ {p}=-2 \end{aligned}$ Since ${p}={3x+5}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${3x+5}=-3\ \ \ \text{or} \ \ \ {3x+5}=-2$ When we solve $3x+5=-3$, we find that $x=-\dfrac{8}{3}$. When we solve $3x+5=-2$, we find that $x=-\dfrac{7}{3}$. In conclusion, the two solutions of the equation $(3x+5)^2+5(3x+5)+6=0$ are $x=-\dfrac{8}{3}$ and $x=-\dfrac{7}{3}$. [Is there another way to solve for x?]